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10-1.Circle and System of Circles
hard
If the circles ${x^2} + {y^2} + 2ax + cy + a = 0$ and ${x^2} + {y^2} - 3ax + dy - 1 = 0$ intersect in two distinct points $P$ and $Q$ then the line $5x + by - a = 0$ passes through $P$ and $Q$ for
A
Infinitely many values of $a$
B
Exactly two values of $a$
C
Exactly one value of $a$
D
No value of $a$
(AIEEE-2005)
Solution
(d) Equation of line $PQ$ (i.e., $common\, chord$) is
$5ax + (c – d)y + a + 1 = 0$…..$(i)$
Also given equation of line $PQ$ is
$5x + by – a = 0$…..$(ii)$
Therefore $\frac{{5a}}{5} = \frac{{c – d}}{b} = \frac{{a + 1}}{{ – a}};$ As $\frac{{a + 1}}{{ – a}} = a$
==> ${a^2} + a + 1 = 0$
Therefore no real value of a exists, (as $D<0$).
Standard 11
Mathematics
